Identifying nondominated alternatives with partial information for multiple-objective discrete and linear programming problems

The problem addressed is that of reducing the set of finite (discrete) multiple criteria alternatives to a subset of alternatives based on three assumptions: (1) the (multiattribute) utility function is additive over attributes, (2) single-attribute utility functions are known, and (3) scaling constants are not known exactly but are specified by a set of linear equalities through interactions with the decisionmaker (DM). The paper develops definitions, theories, and computationally efficient procedures to determine whether an alternative is worthy of further consideration or should be eliminated or is the most preferred alternative with respect to the given partial information on the utility function. The concepts of convex and trade-off nondominancy are defined, and their relation to utility nondominancy is established. All ensuing problems can be solved by linear programming. A computationally efficient algorithm is discussed. The concepts developed are also useful for interactive approaches for selecting nondominated alternatives, building partial information on the DM’s preferences, and obtaining the most preferred alternative with minimal interaction. The minimum partial information needed to establish the most preferred alternative is identified. It is shown that paired comparison questions can be used to assess such minimum partial information. An interactive approach is outlined to obtain the most preferred alternative. It is shown that multiattribute discrete problems can be formulated as multiple objective linear programming (MOLP) problems. All the concepts and procedures developed for discrete sets are extended for MOLP problems. It is shown that all results can be directly extended for quasi- (weak) nondominancy and the new concept of reference nondominancy.